Contour lines of the two-dimensional discrete Gaussian free field

• 成果类型: Article
• 署名作者: Schramm, Oded; Sheffield, Scott
• 署名单位: Microsoft; New York University
• 刊物名称: ACTA MATHEMATICA
• ISSN/ISSBN: 0001-5962
• DOI: 10.1007/s11511-009-0034-y
• 发表日期: 2009
• 页码: 21-137
• 关键词: brownian intersection exponents; erased random-walks; conformal-invariance; critical percolation; coulomb gas; models; VALUES; plane; limit

摘要

We prove that the chordal contour lines of the discrete Gaussian free field converge to forms of SLE(4). Specifically, there is a constant lambda > 0 such that when h is an interpolation of the discrete Gaussian free field on a Jordan domain-with boundary values -lambda on one boundary arc and lambda on the complementary arc-the zero level line of h joining the endpoints of these arcs converges to SLE(4) as the domain grows larger. If instead the boundary values are -a < 0 on the first arc and b > 0 on the complementary arc, then the convergence is to SLE(4; a/lambda - 1, b/lambda - 1), a variant of SLE(4).